Problem 13 - A Graphical Approach to Precalculus with Limits

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$\frac{x^{2}}{4}=1-\frac{y^{2}}{9}$$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Rewrite the Equation

The given equation is \( \frac{x^{2}}{4} = 1 - \frac{y^{2}}{9} \). First, let's rearrange it to recognize its form. Add \( \frac{y^2}{9} \) to both sides to obtain \( \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \).

Step 2: Identify the Equation Form

The equation \( \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \) is in the form of \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This is the standard form of an ellipse equation, where both coefficients of \( x^2 \) and \( y^2 \) are positive, and there is an equal sign with 1 on the right side.

Key Concepts

Ellipse

An ellipse is a special type of conic section, which you can think of as a stretched circle. Unlike a circle where all points are equidistant from the center, an ellipse has two distinct focal points. The sum of the distances from any point on the ellipse to these two foci is always constant. This unique property gives the ellipse its elongated shape.

• Every ellipse has a major axis, which is the longest diameter.
• It also has a minor axis, which is the shortest diameter.

In the previous exercise, the equation we worked with, when rewritten, displayed the characteristics of an ellipse. This is identified by terms involving both \( x^2 \) and \( y^2 \) being squared and set over distinct denominators. The equation ultimately resembles the classic form or pattern of an ellipse.

Standard Form

The standard form of an ellipse's equation helps us easily identify its characteristics and important attributes. It is generally presented as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) or \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), depending on orientation.

• When \( a > b \), the ellipse is horizontally oriented.
• Conversely, when \( b > a \), the ellipse is vertically oriented.

The numbers \( a^2 \) and \( b^2 \) are key to finding the lengths of the major and minor axes of the ellipse. The square roots of these values (\( a \) and \( b \)) are the semi-major and semi-minor axes, respectively. Recognizing an equation in this standard form as we did in the exercise simplifies the process of identifying its graph as an ellipse.

Equation Rearrangement

Rearranging equations is a crucial step in transforming them into easily recognizable forms. This process typically involves moving terms, adding, subtracting, multiplying, and dividing both sides of the equation to change its presentation without altering its inherent properties.

• In solving the original exercise, we added \( \frac{y^2}{9} \) to both sides to achieve a positive form.
• This allowed us to clearly identify the equation as an ellipse because it matched the standard form.

Rearrangement is important, not just for simplifying terms but also for leveraging mathematical patterns to better understand equations. Once the equation is in a standard format, it becomes much easier to interpret the type of conic section it represents. This clarity unlocks deeper insights into its geometry and other properties.